Combined Trefftz methods of particular and fundamental solutions for corner and crack singularity of linear elastostatics

Lee, Ming‐Gong; Young, Lih-Jier; Li, Zi‐Cai; Chu, Po-Chun

doi:10.1016/j.enganabound.2010.02.005

Cited by 19 publications

(29 citation statements)

References 61 publications

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“…Some trial computations in [15] display that the Trefftz methods using (7.5) and (7.6) makes the same performance as those using (6.15) and (6.16). For the HTM, we apply Lagrange multiplier ξ coupling the traction conditions on BC only.…”

Section: Numerical Examplesmentioning

confidence: 69%

“…Hence the admissible functions exactly satisfy the Cauchy-Navier equations. For corner singularity, the singular solutions and their numerical algorithms are explored in [15,18]. Based on the minimum principle of complementary strain energy [5], when the traction condition (3.2) is first enforced, the displacement condition (3.1) is a natural consequence.…”

Section: Hybrid Trefftz Methodsmentioning

confidence: 99%

“…It is proven in [15] that the principal FS between (6.15) & (6.16) with c i = 0 and the complex fundamental solutions (CFS) given in [22,23] are equivalent to each other. By trial computation [15], when the simple fundamental solutions ∇(− ln r i ) are ignored, the accuracy and stability of the CTM remain nearly the same.…”

Section: A Description Of Problemsmentioning

confidence: 96%

“…When there exists a corner singularity, the singular solutions are provided in [15,18]. In this article, we only consider the smooth solutions.…”

Section: Error Analysis Of Ctm and Htm Using Particular Solutionsmentioning

confidence: 98%

“…Since no multiplier is involved in CTM, the simplicity of algorithms and programming grants the CTM a remarkable advantage for elasticity and other complicated problems. Also the numerical computation displays that using PS is more efficient than using FS in both HTM and CTM (see [15]). The comparisons of HTM and CTM using fundamental solutions (FS) and PS are the next aim of this article.…”

Section: Introductionmentioning

confidence: 97%

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Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

2011

Numerical Methods Partial

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For linear elastostatics, the Lagrange multiplier to couple the displacement (i.e., Dirichlet) condition is well known in mathematics community, but the Lagrange multiplier to couple the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the Hybrid Trefftz method (HTM). However, there has not been any analysis for these Lagrange multipliers to couple the traction condition so far. New error analysis of the HTM for elasticity problems is explored in this paper, to derive error bounds with the optimal convergence rates. Numerical experiments are reported to support this analysis. The error analysis of the HTM for linear elastostatics is the main aim of this paper. In this paper, the collocation Trefftz method (CTM) without a multiplier is also introduced, accompanied with error analysis. Numerical comparisons are made for HTM and CTM using fundamental solutions (FS) and particular solutions (PS). The error analysis and numerical computations show that the accuracy of the HTM is equivalent to that of the CTM, but the stability of the CTM is good. For elasticity and other complicated problems, the simplicity of algorithms and programming grants the CTM a remarkable advantage. More numerical comparisons show that using PS is more efficient than using FS in both HTM and CTM. However, since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM. The comparisons of HTM and CTM using FS and PS are the next aim of this article.

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Section: Numerical Examplesmentioning

confidence: 69%

Section: Hybrid Trefftz Methodsmentioning

confidence: 99%

Section: A Description Of Problemsmentioning

confidence: 96%

“…When there exists a corner singularity, the singular solutions are provided in [15,18]. In this article, we only consider the smooth solutions.…”

Section: Error Analysis Of Ctm and Htm Using Particular Solutionsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 97%

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Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

2011

Numerical Methods Partial

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Collocation Trefftz methods for the stokes equations with singularity

Li¹,

Lee²,

Chiang³

2012

Numerical Methods Partial

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This study explores the boundary methods for the two‐dimensional homogeneous Stokes equations and investigates the particular solutions (PS) satisfying the Stokes equations. Smooth solutions for the Stokes equations are provided by explicit fundamental solutions (FS) and PS in this study, and singular corner solutions are also provided from linear elastostatics given in Li et al. ([Eng. Anal. Bound. Elem. 34 (2009), 533‐648, 2009). A new singularity model with an interior crack is proposed and solved by the collocation Trefftz method (CTM). The proposed method achieves highly accurate solutions with the first leading coefficient having 10 significant digits. These solutions may be used as a benchmark for testing results obtained by other numerical methods. Error bounds are derived for the CTM solutions using the PS. For a general corner, the exponent νk in r can only be obtained by numerical solutions of a system of nonlinear algebraic equations. Therefore, the combined method using many FS plus a few singular solutions is inevitable in most applications. For singularity problems, combining a few singular solutions with the FS is an advanced topic and is successfully implemented in Lee et al. (Eng. Anal. Bound. Elem. 24 (2010), 632–654); however, combining a few singular solutions with the smooth PS fails to converge in the first leading coefficient. As a result, the aforementioned method is not applicable to the singularity problems addressed in this article. With the help of particular and singular solutions, the hybrid Trefftz method with Lagrange multipliers can be developed for the Stokes equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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Solution of the first biharmonic problem by the Trefftz method

Jaworski

2013

Z Angew Math Mech

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The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical and follows Mikhlin [29] approach to the solution of the harmonic problem. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges.

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Combined Trefftz methods of particular and fundamental solutions for corner and crack singularity of linear elastostatics

Cited by 19 publications

References 61 publications

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

Collocation Trefftz methods for the stokes equations with singularity

Solution of the first biharmonic problem by the Trefftz method

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